I'm wondering whether the fundamental group of a point and a sphere are the same? If it is, then the topological of a point and a sphere are the same? and we can deform a sphere to a point and still keep the topology?
Would like to know more formal answer or a proof.
Yes. They're both simply connected, assumed you are asking for the fundamental group and the 2-sphere $S^2$.
Absolutely no. You're very far from the properties required to two spaces to be homeomorphic: even if they had the same homotopy type that would not be true. For this specific instance the easiest way to prove that they're not homeomorphic is that they're obviously not bijective.
A more friendly counterexample to the general statement that if two spaces are homotopy equivalent then they are homeomorphic is this: take $X = \mathbb R$ with the usual topology and $Y = \left[ 0,1 \right]$ with the induced topology. They're both contractible, but $Y$ is compact, while $X$ isn't.
The answer is again no. There's no way to deform a sphere to a point: the sphere is not contractible. For showing this the fundamental group is not sufficient; you should look at the second group of homotopy $\pi^2$. Instead of showing that $\pi^2(S^2) = \mathbb Z$, which is not really immediate, I'll give you an example to understand the intuition behind this result.
When we say deformation we really mean that it exists a continuous function from $S^2 \times I \to \{ p \}$, that we can imagine as a deformation over time of $S^2$. The point is, this deformation has to be continuous, and that means in a certain sense we can't generate or eliminate holes. Now, $S^2$ contains an empty 2-dimensional hole, and there's no way to keep this hole while stretching $S^2$ to a point: sooner or later it must vanish; and that can't be done from a continuous function.